Guided NotesChapter 2

Functions, Equations, and GraphsAnswer Key

Unit Essential QuestionsDoes it matter which form of a linear equation that you use?

How do you use transformations to help graph absolute value functions?

How can you model data with linear equations?

2.1 Part 1: Relations and Functions Students will be able to graph relations.

Students will be able to identify functions.Warm Up

Graph each ordered pair on the coordinate plane.

1. (-4, -8) 2. (3, 6) 3. (0, 0) 4. (-1, 3)

Key Concepts

Relation - a set of pairs of input and output values.

Examples1. When skydivers jump out of an airplane, they experience free fall. At 0 seconds,

they are at 10,000ft, 8 seconds, they are at 8976ft, 12 seconds, they are at 7696ft, and 16 seconds, they are at 5904ft. How can you represent this relation in four different ways?

Key Concepts

Domain - the set of all inputs (x-coordinates)Range - the set of all outputs (y-coordinates)Function - a relation in which each element in the domain corresponds to exactly one element of the range.

Functions, Equations, and Inequalities – Algebra 2 2

Vertical Line Test - if any vertical line passes through more than one point on the graph of a relation, then it is not a function.

Examples2. Determine whether each relation is a function. State the domain and range.

a. {(0, 1), (1, 0), (2, 1), (3, 1), (4, 2)}Domain: {0, 1, 2, 3, 4} Yes, it is a function.Range: {0, 1, 2}

b. {(1, 4), (3, 2), (5, 2), (1, -8), (6, 7)}Domain: {1, 3, 5, 6} No, it is not a function.Range: {-8, 2, 4, 7}

c. {(1, 3), (2, 3), (3, 3), (4, 3), (5, 3)}Domain: {1, 2, 3, 4, 5} Yes, it is a function.Range: {3}

d. {(4, 9), (4, 3), (4, 0), (4, 4), (4, 1)}Domain: {4} No, it is not a function.Range: {0, 1, 3, 4, 9}

3. Use the vertical line test. Which graphs represent a function?

Section 2.1 Part 2: Relations and FunctionsStudents will be able to write and evaluate functions

Warm Up

1. Can you have a relation that is not a function?

yes, a relation is any set of pairs of input and output values.

2. Can you have a function that is not a relation?

no, a function is a relation.

Key ConceptsFunction Rule - an equation that represents an output value in terms of an input value. You can write the function rule in function notation.Independent Variable - x, represents the input value.

Functions, Equations, and Inequalities – Algebra 2 3

Dependent Variable - y, represents the output value. (Call dependent because it depends on the input value)

Examples1. Evaluate each function for the given value of x, and write the input x and output f (x)

in an ordered pair.f (x) = -2x + 11 for x = 5, -3, and 0

f (5) = 1; (5, 1) f (-3) = 17;(-3, 17) f (0) = 11; (0, 11)

2. Write a function rule to model the cost per month of a long-distance cell phone calling plan. Then evaluate the function for given number of minutes.

Monthly service fee: $4.99Rate per minute: $.10Minutes used: 250 minutes

2.2: Direct Variation Students will be able to write and interpret direct variation equations

Warm Up

Solve each equation for y.

1. 2. 3.

Key Concepts

Direct Variation - a linear function defined by an equation of the form y=kx, where k ≠ 0.

Constant of Variation - k, where k = y/x

Examples1. For each function, determine whether y varies directly with x. If so, find the

constant of variation and write the equation.

a. b. Functions, Equations, and Inequalities – Algebra 2 4

Does not vary directly k = 2; y = 2x

2. For each function, tell whether y varies directly with x. If so, find the constant of variation.

a. 3y = 7x + 7 b. 5x = –2y

3. Suppose y varies directly with x, and y = 15 when x = 27. Find y when x = 18.

2.3: Linear Functions and Slope-Intercept Form Students will be able to graph linear equationsStudents will be able to write equations of line

Warm UpEvaluate each expression for x = –2, and 0.

1. f(x) = 2x + 7 2. f(x) = 3x – 2

Key ConceptsSlope - the rate of change

POSITIVE NEGATIVE ZERO UNDEFINED

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Linear Function - a function whose graph is a lineLinear Equation - represents a linear function where a solution is any ordered pair (x,y) that makes the equation true.x -intercept - the point in which a line crosses the y-axisy -intercept - the point in which a line crosses the x-axis

Slope Intercept Form y = mx + b

m = slope; (0, b) = y-intercept

Examples1. Find the slope of the line through the points:

a. (3, 0) and (5, 8) b. (1, –4) and (2, –5) c. (–2, 7) and (8, –6)

4 -1 -13/10

2. What is an equation of the line that has a slope of 1/5 and the y-intercept is (0, -3)?

3. Write the equation in slope-intercept form and then find the slope and y-intercept of –7x + 2y = 8.

Slope = 7/2, y-intercept = (0, 4)

Functions, Equations, and Inequalities – Algebra 2 6

4. Graph the equation y = 2x + 1.

2.4 Part 1: More about Linear Equations Students will be able to write equations of lines

Warm UpEvaluate for x = 0.

1. 5x + 2 2. (x - 4) + 12 3. 13 - 6.5x

2 8 13

Key Concepts

Point-Slope Form

(y-y1) = m(x-x1)

Use this form when you are given a point (x1, y1) and the slope (m).

Examples1. A line passes through (-4,1) with slope 2/5. What is the equation of the line in point-

slope form?

2. Write in point-slope form an equation of the line through (4, –3) and (5, –1).

Key Concepts

Standard Form of a Linear Equation

Ax + By = C

Functions, Equations, and Inequalities – Algebra 2 7

where A, B, and C are real numbers, and A and B are not both zero.

Steps to writing an equation in standard form:1. Multiply each term to clear fractions or decimals2. Isolate the variables on the left side of the equation

Examples3. Write each equation in standard form. Use integer coefficients.

a. y = 3/4x - 5 b. y = -4.2x - 5.5

2.4 Part 2: More about Linear Equations Students will be able to write and graph the equation of a line

Students will be able to write equations of parallel and perpendicular linesWarm Up

Find the slope through the following points.1. (2, 4) and (-1, 5) 2. (-7, -8) and (-2, -8)

-1/3 0

Key ConceptsParallel Lines - the slopes of parallel lines are equal. m1 = m2

Perpendicular Lines - the slopes of perpendicular lines are negative reciprocals of each other.

m1 = - 1/m2 Examples

1. What are the intercepts of 3x + 5y = 15? Graph the equation.

(5, 0) and (0, 3)

2. Write in slope-intercept form an equation of the line through (1, –3) and parallel to y = 6x - 2. Graph to check your answer.

3. Write in slope-intercept form an equation of the line through (8, 5) and perpendicular to y = -4x + 6. Graph to check your answer.

Functions, Equations, and Inequalities – Algebra 2 8

2.6 Part 1: Families of Functions Students will be able to analyze transformations of functions

Warm UpGraph each pair of functions on the same coordinate plane.

1) y = -x 2) y = x + 1 3) y = -1/4x y = x y = 2x - 1 y = -1/4x + 2

Key ConceptsParent Function - the simplest form in a set of functions that form a familyTranslation - operation that shifts a graph horizontally, vertically, or both.Vertical Translation - moves the graph up or down, represented by k.

y = f(x) ± kAddition moves it up, subtraction moves it down. (Same Sign)

Horizontal Translation - moves the graph right or left, represented by h.y = f(x ± h)

Addition moves it left, subtraction moves it right. (Opposite Sign)

Examples1. Describe how the functions y = 2x and y = 2x + 3 are related. How are their graphs

related?

y = 2x + 3 is a vertical shift up 3 units of the parent function y = 2x.

Graphs are both linear and parallel.

Functions, Equations, and Inequalities – Algebra 2 9

2. Make a table of values for f(x) = x2 and then for g(x) = (x – 5)2. Describe the transformation.

g(x) is shifted horizontally 5 units to the right from f(x).

3. Write an equation to translate the graph of:a. y = 4x, 5 units down. b. y = 6x, 3 units to the right.

4. Write an equation for each translation of y = x2.a. 3 units up, 7 units right

b. 5 units down, 1 unit left

2.6 Part 2: Families of Functions Students will be able to analyze transformations of functions

Warm UpGraph each pair of functions on the same coordinate plane. 1. y = x, y = x + 4 2. y = –2x, y = –2x – 3 3. y = x, y = x – 2

Functions, Equations, and Inequalities – Algebra 2 10

Key ConceptsReflection - flips the graph of a function across a line such as the x- or y-axis.*For the function f(x), the reflection in the y-axis is f(-x).*For the function f(x), the reflection in the x-axis is –f(x)

Examples1. Write the function rule for each function reflected in the given axis.

a. f(x) = 2x – 1 x-axis: y = -2x + 1y-axis: y = -2x - 1

b. f(x) = x + 3 x-axis: y = -x – 3y-axis: y = -x + 3

Key Concepts

Vertical Stretch - multiplies all y-values of a function by the same factor greater than 1.

Vertical Compression - reduces all y-values of a function by the same factor between 0 and 1.In other words, let f (x) be the parent function,

if a > 1 then y = af(x) is a vertical stretch if 0 < a < 1 then y = af(x) is a vertical compression

Examples2. Graph each of the following on the same coordinate plane and create a table of

values for each. Explain the transformations from the first equation.

Functions, Equations, and Inequalities – Algebra 2 11

Stretch by 3; Compression 1/3

3. The graph of g(x) is the graph of f(x) = 6x compressed vertically by the factor 1/2 and then reflected in the y-axis. What is the function rule for g(x)?

4. What transformations change the graph of f(x) to the graph of g(x)?f(x) = 2x2 g(x) = 6x2 – 1

Stretch by 3 unitsVertical shift down 1 unit

Concept Summary

2.7: Absolute Value Functions and Graphs

Students will be able to graph absolute value functionsWarm Up

Solve each absolute value equation.

1. 2.

Functions, Equations, and Inequalities – Algebra 2 12

x = 8, -2 x = 21/5, -3

Key Concepts

Graph of Absolute Value Function

Axis of Symmetry - the line that divides a figure into two parts that are mirror images

Vertex - a point where the function reaches a maximum or minimum value

Examples1. What is the graph of the absolute value function y = |x| - 4

Key Concepts

Functions, Equations, and Inequalities – Algebra 2 13

Examples2. Graph each absolute value function.

a. y = |x + 2| + 3 b. y = 1/2|x|

Key Concepts

General Form of the Absolute Value Function

y = a |x-h| + k

Stretch/Compression Factor is |a|, Vertex is (h, k), Axis of Symmetry is x = h

Examples

3. Without graphing, identify the vertex, axis of symmetry, and transformations of f(x) = -3|x – 1| + 4 from the parent function f(x) = |x|.

Vertex: (1, 4), Axis of Symmetry: x = 1Stretch by 3 units, Reflect over the x-axis, Horizontal shift right 1 unit, Vertical shift up 4 units

4. Write an absolute value equation for the given graph.

2.8: Two Variable InequalitiesStudents will be able to graph two-variable inequalities

Warm UpSolve each inequality. Graph the solution on a number line.1. 2. 3.

p ≤ 5/4 t > 13 t ≤ -3

Key ConceptsLinear Inequality -an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.

Functions, Equations, and Inequalities – Algebra 2 14

Steps to graph two-variable inequality:1. Graph the boundary line (graph as if it was an equation)

If y< or y>, then the boundary is dashed. If y≥ or y≤, then the boundary is solid.

2. Shade the solutions For y< or y≤, shade below the boundary. For y> or y≥, shade above the

boundary.

Examples1. Graph

2. Graph

3. Graph

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